UV Editor: unwrapped faces disappear if I select other faces
There are two things going on here:


When you hit U > Unwrap, Blender only unwraps the selected geometry.  To unwrap the head and tail of the mushroom, select them both before unwrapping.
The UV/Image Editor view only shows the pelt[s] for geometry that is selected in the 3D View.  When you deselect something in the 3D view, it's unwrapped texture coordinates aren't deleted, they are just no longer visible in the UV/Image Editor.You can optionally use Sync Selection option so the UV selection matches the 3D view-port. This can be toggled using the button to the right hand side of pivot point popup.